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A brief explanation of why primes peak at repeated multiples from a layman who's wondered why before. Obvious first example all primes above 2 are of the form
by AGoodName 8y ago
A brief explanation of why primes peak at repeated multiples from a layman who's wondered why before.
Obvious first example all primes above 2 are of the form 2x+1. An obvious repeating pattern of primes.
You can take this a small step further. All primes above 6 are of the form 6x+1 or 6x+5. Anything else is a multiple of 2 or 3. Above 6 only 1/3 of numbers are worthy of being considered prime. This is a slightly less obvious example.
A small step further - all primes above 30 are of the form 30x+1, 30x+7, 30x+11, 30x+13, 30x+17, 30x+19, 30x+23 or 30x+29. Anything else is a multiple of 2,3 or 5. So above 30 only 8/30 numbers are worthy of being considered prime. See how we've created a new pattern for the multiple of 2x3x5 to rule out a swath of prime candidates..
I could repeat this each prime found. eg. I could take the common multiple of 2,3,5,7 (210) and create a similar pattern for all numbers above 210 that rules out the repeated multiples of 2,3,5 and 7. (leaving us just 58/210 numbers worthy of being considered prime).
This is why you see peaks of primes at various repeating multiples. For every new prime found you can take the multiple of it and all previous primes. From that you can rule out primality for various offsets to any multiples of that number. So primes above certain numbers can only possibly exist in certain forms. Which is why you see primes at repeated patterns from each other - the primes can only exist in those forms.
- taneq 8y agoThat seems so obvious when you put it like that. (Although I guess all of mathematics is either "obvious" or "unsolved".)
- ipsin 8y agoI think most of what we've solved in math is not obvious. Some accessible examples would be Fermat's Last Theorem or the four color problem.
- deleted 8y ago[deleted]
- twanvl 8y agoSmall correction: there are only 48 (not 58) numbers out of 210 that are not multiples of 2,3,5,7.
- chongli 8y agoThis pattern forms the basis for wheel factorization [1], a faster way to factor a number than naïve trial division. [1] https://en.wikipedia.org/wiki/Wheel_factorization https://en.wikipedia.org/wiki/Wheel_factorization
- akira2501 8y agoAlso, the https://en.wikipedia.org/wiki/Sieve_of_Atkin https://en.wikipedia.org/wiki/Sieve_of_Atkin
- AGoodName 8y agoAuthor of this thread here. That article states that the above sieve was created in 2003. I actually wrote and described it back in 2002 - https://forums.overclockers.com.au/threads/show-off.71630/#post-813807 https://forums.overclockers.com.au/threads/show-off.71630/#p... I didn't think much of it back when i did it and i'm not a mathematician, just a hobbyist. I guess i should publish my prime hobbies more...
- gtt 8y agoHave you though about mailing it to authors of the paper? The should somehow cite you, I guess.
- s-shellfish 8y agoI haven't done all the math for this (I've deeply investigated the pattern for 2x+1) but it seems like this would be an obvious and intuitive result of primes. You are still generating primes from primes. Yes, you find more primes, but the computation is still dependent on primes. I'm still of the opinion that there is no complete pattern to the primes. I'm assuming the researchers do not have the intent of confusing a crystal lattice structure with an actual mathematical lattice, because while they possibly may share similar influences in their models, one is math, the other is physics. #keepmathpure
- calibas 8y agoI've heard this sentiment before and I don't really understand it. There's no separating physics and math. Keeping math "pure" really means keeping it "purely abstract", so it resists any kind of practical application.
- s-shellfish 8y agoI don't agree with this. And honestly, I really think that depends on what foundation you rely on to think with, work with, create with, test with, and check your own tests with. Physics does not have to use itself to understand itself. Math does.
- Numer 8y ago> Physics does not have to use itself to understand itself. Could you explain what you mean by that?
- s-shellfish 8y agoIt's simple. Physics uses mathematics to construct equations that describe properties of physics. The difference is physics has reality to test against - observations that can be measured. Math does not have this. Math's only metric against itself is itself.
- throwawaymath 8y ago
- sxv 8y agoIn your example you started at 1/3 (33.33%) and moved to 58/210 (27.62%). What does this limit approach ad infinitum?
- hiq 8y agoYou can have a look at https://en.wikipedia.org/wiki/Prime_number_theorem https://en.wikipedia.org/wiki/Prime_number_theorem